Problem 59 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$7 x-\frac{12}{x}$$

Step-by-Step Solution

Answer

The value of the expression is \(-25\).

Step 1: Identify the Expression

The given expression to evaluate is \(7x - \frac{12}{x}\). We need to substitute \(x = -4\) into this expression.

Step 2: Substitute the Value

Replace \(x\) with \(-4\) in the expression. This gives us: \(7(-4) - \frac{12}{-4}\).

Step 3: Calculate Product and Quotient

First, calculate \(7 \times (-4) = -28\). Then compute the division \(\frac{12}{-4} = -3\).

Step 4: Combine Results

Subtract the result of the division from the product: \(-28 - (-3)\). This simplifies to \(-28 + 3\).

Step 5: Finalize the Calculation

Perform the final arithmetic operation: \(-28 + 3 = -25\).

Key Concepts

Replace Variable

When tasked with evaluating an expression, an essential first step is to replace any variables within the expression with given values. This process is also known as substitution. In the context of our exercise, you are given the expression \(7x - \frac{12}{x}\) and told to replace \(x\) with \(-4\). This means wherever you see the variable \(x\), you substitute it with the number \(-4\).

Thus, the expression \(7x - \frac{12}{x}\) becomes \(7(-4) - \frac{12}{-4}\). Doing this correctly allows you to proceed with next steps in solving the expression, now free of variables.

Arithmetic Operations

Once you have replaced the variables, the next task is to perform arithmetic operations in proper order. In this example, we have a multiplication and a division to carry out.

• First, multiply the coefficient \(7\) by the replaced variable \(-4\). This gives us \(7 \times (-4) = -28\).
• Next, divide \(12\) by \(-4\) to get the result \(-3\). This is done by taking \(\frac{12}{-4} = -3\).

Handling operations in a systematic and ordered manner ensures that each part is simplified correctly, aligning with the rules of arithmetic.

Negative Numbers

Working with negative numbers requires careful attention to the rules of arithmetic involving negatives. A common area of confusion can arise when subtracting negative numbers, but understanding how negatives interact can simplify your calculations.

Using the results from our previous steps, our expression simplifies to \(-28 - (-3)\). An important rule when dealing with subtraction of a negative is that subtracting a negative number is the same as adding its positive. Therefore, \(-28 - (-3)\) is equivalent to \(-28 + 3\).

Ultimately, handling negative numbers correctly is crucial to arriving at the correct answer. For this exercise, after performing \(-28 + 3\), you obtain the result \(-25\). This final result reflects a proper understanding of negative arithmetic operations.